Estimation of the Normal Contact Stiffness for Frictional Interface In

Article Estimation of the Normal Contact Stiﬀness for Frictional Interface in Sticking and Sliding Conditions

Davide Tonazzi 1 , Francesco Massi 1 , Mario Salipante 1, Laurent Baillet 2 and Yves Berthier 3 1 DIMA, Department of Mechanical and Aerospace Engineering, “La Sapienza” University of Rome, via Eudossiana 18, 00184 Rome, Italy 2 Grenoble Alpes University, CNRS, IRD, IFSTTAR, ISTerre, 38000 Grenoble, France 3 LaMCoS, Contacts and Structural Mechanics Laboratory, University of Lyon, CNRS, INSA Lyon, UMR 5259, 20 rue des Sciences, F-69621 Villeurbanne, France * Correspondence: [email protected]

Received: 12 June 2019; Accepted: 23 June 2019; Published: 2 July 2019

Abstract: Modeling of frictional contact systems with high accuracy needs the knowledge of several contact parameters, which are mainly related to the local phenomena at the contact interfaces and affect the complex dynamics of mechanical systems in a prominent way. This work presents a newer approach for identifying reliable values of the normal contact stiffness between surfaces in contact, in both sliding and sticking conditions. The combination of experimental tests, on a dedicated set-up, with finite element modeling, allowed for an indirect determination of the normal contact stiffness. The stiffness was found to increase with increasing contact pressure and decreasing roughness, while the evolution of surface topography and third-body rheology affected the contact stiffness when sliding.

Keywords: contact stiﬀness; friction; contact dynamics; numerical tribology

1. Introduction The correct modeling of contact interfaces is a key factor in many engineering applications that inherently include contacts between components, e.g., friction joints [1–3], clamping devices, robotic contacts, rolling bearings [4,5], brake systems [6], etc. Contact mechanics between component surfaces also play a key role for understanding many tribological processes, such as friction-induced vibrations [7–11], stick-slip phenomena [12–14], frictional instability [7,15], adhesion, and wear [16,17]. An accurate contact modeling of such systems requires knowledge of interface contact parameters [18] such as contact stiffness, which are not easily measurable and completely understood. In the last decades, numerical simulation became one of the most powerful tools for both the design of complex mechanical systems and the investigation of dynamic or tribological issues in dry contacts [16,19]. On one hand, industries use common numerical finite element models for the estimation of stress and strain distributions by static analysis [4,5], eigenvalues, and eigenmodes by modal analyses [6,20,21], or dynamic response of full systems through transient simulations. On the other hand, researchers have drastically improved the capability of numerical codes for understanding and solving any kind of issue related to contact interfaces and systems under frictional contacts. For example, transient and modal numerical analyses are nowadays used for predicting and simulating contact-induced instabilities, such as brake squeal [6,22,23], creep groan noise [24], or stick–slip instabilities [15,25,26]. The development of more and more sophisticated and refined numerical models [27,28] of systems with contact interfaces, requires us to define more reliable contact parameters that could largely affect

Lubricants 2019, 7, 56; doi:10.3390/lubricants7070056 www.mdpi.com/journal/lubricants Lubricants 2019, 7, 56 2 of 17 both the tribological and dynamic response of the system [29]. With regards to this, this work was focused on the estimation of normal contact stiffness at a frictional interface. The first studies dealing with contact stiffness used analytical approaches through simple elastic models, such as Hertzian spherical contact. In the GW (Greenwood and Williamson) model [30], the contact interface is considered nominally flat with a single scale roughness, where the contact stiffness is obtained by a statistical model. Starting with the pioneering work of Greenwood and Williamson, various analytical and multiscale models [31] of contact interfaces have been developed, which focus on the effect of roughness on the normal and tangential contact behavior [32,33]. The effect of roughness on the normal and tangential contact stiffness and energy dissipation in frictional contacts has been recently investigated in [34,35], while a comparison between various statistical and fractal models is presented in [36,37]. Some experimental approaches to measure contact stiffness are based on ultrasonic methods, where high frequency waves are used to derive the stiffness from the interface [38]. Another method to measure contact stiffness and damping at the nano-scale has been presented by Asif et al. [39]. A few works [40,41] in the literature are also based on combined experimental and analytical/numerical approaches for indirect identification of the contact stiffness [42]. In this context, the objective of the present work was to estimate contact stiffness by combining experimental dynamic tests and a continuous numerical approach, for an indirect measurement of the stiffness at the contact interface. A specific tribometer has been developed for identifying the contact stiffness in both sticking and sliding conditions, for different surface topographies, materials, geometries, and average contact pressures. On one hand, the frequency analysis of the system response allowed for identifying its dynamics, which was related to both the system and the interface contribution. On the other hand, a finite element model of the experimental set-up was developed by considering the contact interface contribution. A comparison between the numerical and experimental results allowed for the estimation of contact stiffness as a function of the investigated parameters. With respect to other approaches in literature, the following improvements could be underlined: The developed test bench enabled to perform tests both in the sliding and sticking conditions, allowing for investigating the actual surface response during sliding. The lumped 3D finite element model [43], allowed to account for the geometry of the contact surface and the tri-dimensional deformations of the bodies in contact. First, the methodology was validated and the results were compared with the ranges present in the literature. Then, a parametrical experimental campaign, combined with the results of the numerical model, allowed for identifying contact stiffness as a function of different average contact pressures and different surface topographies in the sticking conditions. The results were compared with the stiffness values in classical finite element models. Lastly, the contact stiffness was estimated in the sliding conditions and compared with the one obtained in the sticking condition.

2. Method and Materials

2.1. Description of the Approach The proposed methodology for the estimation of the normal contact stiffness was based on the dynamic contribution of the interface to the dynamic response of the mechanical system. A combined numerical and experimental approach (Figure1) was proposed for an indirect identification of the contact stiffness parameter. From an experimental point of view, a setup was designed and used to perform dynamic tests (Figure2). Afterwards, a 3D finite element model—updated and representative of the experimental setup—was utilized to compare the numerical and experimental results for estimating the normal contact system between the considered contact surfaces. Lubricants 2019, 7, 56 3 of 17 Lubricants 2019, 7, x FOR PEER REVIEW 3 of 17

Figure 1. Overview of the proposed approach for the esti estimationmation of the contact stistiffness.ﬀness. The dynamic responses ofof the the numerical numerical model model and and the experimentalthe experimental system system were comparedwere compared to retrieve to retrieve the stiﬀness the Lubricantscontributionstiffness 2019, 7contribution, x FOR of the PEER contact of REVIEW the interface. contact interface. 5 of 17

The comparison between the results from the numerical model and the frequency response function obtained from the experiments allowed for retrieving the value of the normal contact stiffness. To elaborate, the finite element model of the set-up (Figure 3) was first developed and validated to simulate the same dynamic response (natural frequencies) measured on the tribometer when the tested sample was not in contact with the disc. Once the finite element model was validated, the contact was established and the only unknown was the contact stiffness. The frequency of the investigated mode (Figure 4) was then the output of the model used that was for the identification of the contact stiffness by comparing it with the experiments. In fact, a parametrical numerical modal analysis was performed as a function of contact stiffness, in order to meet the natural frequency of the investigated mode with the one measured experimentally. The numerical value of the contact stiffness that allowed for a matching between experimental and numerical natural frequencies was retained as the estimated value of the interface stiffness. Moreover, to verify that the contact stiffness parameter was decoupled from the stiffness of the bulk, the test was repeated for different sample length, by changing the stiffness of the bulk and leaving the same contact stiffness at the interface.

This procedure was repeated for each investigated parameter—average contact pressure and Figure 2. Scheme (left) and photograph (right) of the developed setup for the contact stiffness roughness.Figure 2. Scheme (left) and photograph (right) of the developed setup for the contact stiffness experiments. Frequency Response Functions were measured to identify the natural frequencies of the experiments.The contact Frequencystiffness was Response introduced Functions in the were finite measured element to identify model the (Figure natural 3) frequencies through a of classical the system, which were directly affected by the contact stiffness. penaltysystem, method which implemented were directly aonffected ANSYS by the (ANSYS, contact stiInc.ffness. Canonsburg, PA, USA), which introduced a stiffness between the nodes of the contact and target surfaces. The introduction of the contact stiffness 2.3. TestThe Sample comparison Definition between the results from the numerical model and the frequency response allowed for a penetration gap between the two surfaces and the corresponding contact force, which function obtained from the experiments allowed for retrieving the value of the normal contact stiffness. was theCylindrical product shapedbetween specimens the contact were stiffness machined and thefor gap.the tests. Samples of polymethyl methacrylate To elaborate, the finite element model of the set-up (Figure3) was first developed and validated to (PMMA)It is wereworth used pointing for the out validation that the introductionof the method ofology. a distributed To achieve surface consistent stiffness results, through it was the simulate the same dynamic response (natural frequencies) measured on the tribometer when the tested necessarypenalty method to define was a protocolvalidated, for by the comparing specimen thepreparation. results with the ones obtained when introducing sample was not in contact with the disc. Once the finite element model was validated, the contact was an interfaceFor each layercylindrical with bar,the correspondingthe geometrical stiffness, and ma terialwhich properties led to the were same measured dynamic in orderstiffness to established and the only unknown was the contact stiffness. The frequency of the investigated mode guaranteecontribution. a strict control on the mechanical and geometrical characteristics of the material. After the (Figure4) was then the output of the model used that was for the identification of the contact sti ffness characterization of the material geometry and mechanical properties (Young Modulus and density), by comparing it with the experiments. the2.2. cylindricalExperimental bars Test (12 Bench mm in diameter) were cut into three pieces of different lengths—10, 15, and 20 mm. The experimental tribometer, designed to perform the experiments for the contact stiffness After the preliminary cutting process, in order to obtain the defined roughness at the contact estimation, is presented in Figure 2. The setup was designed to perform tests with different materials surface, a face milling process was performed on both contact surfaces; using the right combination (samples) in contact, different lengths of the samples, and for a range of the average contact pressure of cutting head (diameter and cutters shape) and rotational and feeding speed, it was possible to up to 1 MPa. The system was mainly characterized by a material sample, which was in contact with obtain a repeatable surface roughness that met the following requirements: a thick steel disc and loaded by the weight of a massive support (guide bar). The surface of the disc Ra (µm) = Ratarget ± 0.2; in contact with the material sample was polished to obtain a value of Ra = 0.1 µm in terms of Slope tolerance (both faces) (‰) = 1;

Flatness tolerance (both faces) (‰) = 1. After the milling process, each specimen was cleaned in an ultrasound cleaning machine and then left to dry. At this time, every single cylinder was visually inspected and tested with a profilometer, in order to evaluate the surface topography. Only the specimens that met the roughness and planarity requirements were used for the contact stiffness evaluation campaign.

2.4.3. D Finite Element Model The finite element model (Figure 3) of the experimental setup was developed using the ANSYS software. In order to reduce the model size and the complexity of the system, the numerical model only included the disc assembly and the air bearing support with its guide bar. The presence of the shaft and bearings with the related global dynamic contributions were taken into account by adding an elastic support on the base of the disc assembly, with a stiffness value calculated by updating the model according to the preliminary dynamic tests.

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FigureFigure 3.3. FiniteFinite elementelement modelmodel ((leftleft)) ofof thethe developeddeveloped experimentalexperimental setupsetup andand zoomed-inzoomed-in viewview ofof thethe samplesample inin contactcontact ((rightright).). TheThe contactcontact stistiffnessﬀness parameterparameter waswas introducedintroduced atat thethe contactcontact interfacesinterfaces ofof Lubricantsthethe tested2019tested, 7, sample. sample.x FOR PEER REVIEW 7 of 17

A hexahedral-dominant mesh was used in the numerical model. Each 3D solid element had 16 nodes, for a total of 48 degrees of freedom. The model was composed of four main bodies—the disc, the guide bar, the air bearing, and the specimen. For each body, a linear elastic material law had taken into account with the following parameters: Steel Disc: Young Modulus E = 201 GPa, Poisson coefficient ν = 0.3, and mass density ρ = 7930 kg/m3; Linear air bearing (support and guide bar): Young Modulus E = 70 GPa, Poisson coefficient ν = 0.3, and mass density ρ = 2700 kg/m3; PMMA specimen: Young Modulus E = 4.5 GPa, Poisson coefficient ν = 0.4, and mass density ρ = 1190 kg/m3. A finer mesh was used for the specimen meshing, while a coarser mesh turned out to be sufficient for an accurate description of the other bodies in the frequency range of interest, for a total of 13,884 elements in the model. The contact region between the guide bar and the air bearing was modeled as frictionless and a pure penalty method was used to describe the contact between the specimen surfaces, the disc, and the guide bar (Figure 3). The value of the normal contact stiffness Kcontact (parameter to be estimated) was set to be equal for both contact interfaces of the sample, with the disc and the supporting bar having the same roughness (Figure 3). A fixed support on one side of the air bearing frame was defined,Figure in 4.order Deformed to reproduce shape of the investigated experimental mode mounting. (490 Hz) forA point sample mass heightheight on HH top == 10 of mm the andguide for bar an was usedaverage to account contact for pressurethe dynamic PP == 0.14 effect MPa. of The Thethe mode modedead corresponded correspondedmass on the topto a ofmass-spring mass-spring the guide deformation, bar in the experimental where setup.the springspring waswas thethe combinationcombination of of the the bulk bulk sti stiffnessffness of of the the sample sample in in series, series, with with the the contact contact sti stiffnessffness at theatAn the interfaces. elastic interfaces. support per unit area on the joint between disc and shaft was placed where the disc physically joined the shaft, with a magnitude of 20 N/mm3, to model the stiffness of the shaft bearings 3.assembly. ContactIn fact, Stiffness a parametrical Estimation numerical in the modal Sticking analysis Condition was performed as a function of contact stiffness, in orderThe to numerical meet the natural update frequency process of of each the investigated components mode and then with theof the one whole measured system experimentally. was carried 3.1. Experimental Dynamic Tests Theout numericalin a preliminary value ofphase. the contact The numerical stiffness thatmodel allowed was updated for a matching in order between to meet experimental the experimental and numericalnaturalA parametrical frequencies natural frequencies modalof the analysisexperimental was retained was performed set-up. as the Afterwards, estimated as a function value the of focus of the the different was interface placed sample sti ffonness. thelengths mode and of ofvibration theMoreover, different involving toaverage verify the thatcontactrigid the vertical pressures contact oscillation sti ff(addedness parameterof masses) the mass that was (air were decoupled guide experimentally and from added the stimass)imposed.ffness and of The thethe bulk,resultsstiffness the of testprovided the was numerical repeated by the modal sum for di of ffanalysis erentthe bulk sample (frequency stiffness length, and of by theinterface changing investigated stiffness the sti ffmode) ofness the of weretested the bulk compared sample and leaving(Figure with the4). Theexperiments same developed contact (frequency sti 3Dffness finite at response theelement interface. model function). allowed An extensive to account measurement for the real deformation campaign was of conductedthe sample forand the Thisthe estimation elastic procedure deformation of the was contact repeated of the stiffness disc for as in eachwell the sticking(Figure investigated condition,4). parameter—average using the PMMA material contact samples. pressure and roughness.Three specimens for each length (L = 10 mm, L = 15 mm, L = 20 mm), for two values of the roughnessThe contact parameter stiffness (Ra was= 1 µm introduced and Ra = in6 µm), the finite were element tested according model (Figure to the3) testing through protocol a classical (see penaltySection 2.2), method for recovering implemented the onFRFs ANSYS (Figure (ANSYS, 5). Inc. Canonsburg, PA, USA), which introduced a stiffness between the nodes of the contact and target surfaces. The introduction of the contact stiffness allowed for a penetration gap between the two surfaces and the corresponding contact force, which was the product between the contact stiffness and the gap.

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It is worth pointing out that the introduction of a distributed surface stiffness through the penalty method was validated, by comparing the results with the ones obtained when introducing an interface layer with the corresponding stiffness, which led to the same dynamic stiffness contribution.

2.2. Experimental Test Bench The experimental tribometer, designed to perform the experiments for the contact stiffness estimation, is presented in Figure2. The setup was designed to perform tests with di fferent materials (samples) in contact, different lengths of the samples, and for a range of the average contact pressure up to 1 MPa. The system was mainly characterized by a material sample, which was in contact with a thick steel disc and loaded by the weight of a massive support (guide bar). The surface of the disc in contact with the material sample was polished to obtain a value of Ra = 0.1 µm in terms of roughness, as for the surface in contact with the guide bar. In comparison to the tested material (polymethyl methacrylate (PMMA) properties, such lower values of the roughness, together with the higher value of the component bulk stiffness (Alloy Aluminum and Stainless Steel), allowed for neglecting the contact stiffness contribution of the disc and the guide bar roughness. In order to minimize the friction between the massive support and the frame, along the normal direction to the contact surface, an air guide (NEWWAY Air-Bearings S40-03125, Aston, PA, USA)—based on the air bearings technology—was used. In this way, the massive support was free to move in the normal direction. The experimental setup was equipped with a DC motor and a digital motion controller, making it possible to perform measurements, in both the sticking and sliding conditions, at a controlled rotational speed. In order to perform the dynamic tests, the setup was equipped with the following instruments:

1 instrumented impact hammer (Brüel & Kjær type 8202, Naerum, Denmark); 2 piezoelectric charge accelerometer (Brüel & Kjær type 4393); 3 system acquisition for noise and vibration measurements (PROSIG type P5600, Fareham, UK); 4 charge ampliﬁers (Brüel & Kjær type 2635).

In order to apply this methodology, the experimental setup was designed by considering that the resonance frequency of interest was easily distinguished from the other resonance frequencies of the experimental setup (Figure5). The dynamic response of the system to an impulsive excitation (hammer impact) was recorded by an accelerometer placed at the top of the guide bar (Figure2). The signal of the impulsive force and the system acceleration were recorded by a signal acquisition system with a sampling frequency of 10 kHz. An extensive measurement campaign was conducted for the measurement in the sticking condition. Three specimens for each length and surface roughness were used, considering the following protocol: Once the specimen was placed between the disc and the guide bar, the system was excited by the instrumented hammer with a single impact on the guide bar and the system response was recovered by the accelerometer (Figure5); The recorded signals were used to compute the frequency response function (FRF), in order to identify the natural frequency of the investigated mode (Figure5). These steps were repeated, maintaining the same material sample in place, for each of the following average contact pressure:

0.14 MPa; 0.25 MPa; 0.35 MPa; 0.57 MPa; 1 MPa.

Finally, the test for an average contact pressure of 0.14 MPa was repeated, in order to ensure the repeatability of the whole set of measurements. LubricantsLubricants 20192019,, 77,, 56x FOR PEER REVIEW 86 of 1717

FigureFigure 5.5. (Top)) AccelerationAcceleration andand forceforce signalssignals overover time,time, duedue toto thethe impactimpact excitationexcitation forfor aa singlesingle measurement.measurement. (Bottom)) FRFsFRFs ofof the system for didifferentﬀerent lengthslengths (10,(10, 15, 20 mm) of the tested samples. TestsTests werewere performed with average average contact contact pressure pressure (ACP) (ACP) of of 0.25 0.25 MPa MPa and and surface surface roughness roughness of Ra of Ra= 6= µm.6 µ m.

2.3. TestThe Sample tests were Definition repeated for each of the following average contact pressure values—0.14 MPa, 0.25,Cylindrical 0.35, 0.57, and shaped 1 MPa. specimens After performing were machined the test for with the tests. the highest Samples pressure, of polymethyl a last measurement methacrylate (PMMA)was performed were usedwith the for lowest the validation value of ofthe the average methodology. contact pressure, To achieve in order consistent to verify results, the eventual it was necessarymodifications to define of the a protocolinterface for (e.g., the roughness specimen preparation. plastic deformation) that could modify the natural frequencyFor each of the cylindrical investigated bar, mode. the geometrical and material properties were measured in order to guaranteeFigure a 5 strict shows control the frequency on the mechanical response functions and geometrical computed characteristics for each tested of the height material. of the Aftersamples, the characterizationhighlighting how of the the natural material frequency geometry of and the mechanical investigated properties mode (at (Young about Modulus400–450 Hz) and shifted density), at thethe cylindricallower frequencies, bars (12 mmwithin the diameter) increase wereof the cut sample into three height. pieces Such of a di trendfferent was lengths—10, due to the 15,decrease and 20 of mm. the sampleAfter bulk the stiffness, preliminary when cutting its length process, increased. in order The tofrequency obtain the peak defined at about roughness 150 Hz was at the related contact to surface,a structural a face mode milling of the process set-up was and, performed as expected, on both it did contact not change surfaces; as a using function the rightof the combination heights of the of cuttingtested sample. head (diameter and cutters shape) and rotational and feeding speed, it was possible to obtain a repeatableAfter surfacecomputing roughness the FRFs that as met a function the following of th requirements:e average contact pressure, sample length and sample roughness (as in Figure 5), the frequency evolution of the investigated mode is plotted in Ra (µm) = Ratarget 0.2; ± FigureSlope 6. A tolerance decrease (both in the faces) natural (% frequency) = 1; as a function of the contact pressure was observed for bothFlatness Ra = 1 µm tolerance and Ra (both= 6 µm. faces) The (% decrease) = 1. in frequency was due to the increase of the added mass weightAfter needed the milling for varying process, the each contact specimen pressure was in cleaned the experiments. in an ultrasound cleaning machine and then left to dry. At this time, every single cylinder was visually inspected and tested with a profilometer, in

Lubricants 2019, 7, 56 7 of 17 order to evaluate the surface topography. Only the specimens that met the roughness and planarity requirements were used for the contact stiﬀness evaluation campaign.

2.4. D Finite Element Model The finite element model (Figure3) of the experimental setup was developed using the ANSYS software. In order to reduce the model size and the complexity of the system, the numerical model only included the disc assembly and the air bearing support with its guide bar. The presence of the shaft and bearings with the related global dynamic contributions were taken into account by adding an elastic support on the base of the disc assembly, with a stiffness value calculated by updating the model according to the preliminary dynamic tests. A hexahedral-dominant mesh was used in the numerical model. Each 3D solid element had 16 nodes, for a total of 48 degrees of freedom. The model was composed of four main bodies—the disc, the guide bar, the air bearing, and the specimen. For each body, a linear elastic material law had taken into account with the following parameters: Steel Disc: Young Modulus E = 201 GPa, Poisson coefficient ν = 0.3, and mass density ρ = 7930 kg/m3; Linear air bearing (support and guide bar): Young Modulus E = 70 GPa, Poisson coefficient ν = 0.3, and mass density ρ = 2700 kg/m3; PMMA specimen: Young Modulus E = 4.5 GPa, Poisson coefficient ν = 0.4, and mass density ρ = 1190 kg/m3. A finer mesh was used for the specimen meshing, while a coarser mesh turned out to be sufficient for an accurate description of the other bodies in the frequency range of interest, for a total of 13,884 elements in the model. The contact region between the guide bar and the air bearing was modeled as frictionless and a pure penalty method was used to describe the contact between the specimen surfaces, the disc, and the guide bar (Figure3). The value of the normal contact sti ffness Kcontact (parameter to be estimated) was set to be equal for both contact interfaces of the sample, with the disc and the supporting bar having the same roughness (Figure3). A fixed support on one side of the air bearing frame was defined, in order to reproduce the experimental mounting. A point mass on top of the guide bar was used to account for the dynamic effect of the dead mass on the top of the guide bar in the experimental setup. An elastic support per unit area on the joint between disc and shaft was placed where the disc physically joined the shaft, with a magnitude of 20 N/mm3, to model the stiffness of the shaft bearings assembly. The numerical update process of each components and then of the whole system was carried out in a preliminary phase. The numerical model was updated in order to meet the experimental natural frequencies of the experimental set-up. Afterwards, the focus was placed on the mode of vibration involving the rigid vertical oscillation of the mass (air guide and added mass) and the stiffness provided by the sum of the bulk stiffness and interface stiffness of the tested sample (Figure4). The developed 3D finite element model allowed to account for the real deformation of the sample and the elastic deformation of the disc as well (Figure4).

3. Contact Stiﬀness Estimation in the Sticking Condition

3.1. Experimental Dynamic Tests A parametrical modal analysis was performed as a function of the different sample lengths and of the different average contact pressures (added masses) that were experimentally imposed. The results of the numerical modal analysis (frequency of the investigated mode) were compared with the experiments (frequency response function). An extensive measurement campaign was conducted for the estimation of the contact stiffness in the sticking condition, using the PMMA material samples. Lubricants 2019, 7, 56 8 of 17

Three specimens for each length (L = 10 mm, L = 15 mm, L = 20 mm), for two values of the roughness parameter (Ra = 1 µm and Ra = 6 µm), were tested according to the testing protocol (see Section 2.2), for recovering the FRFs (Figure5). The tests were repeated for each of the following average contact pressure values—0.14 MPa, 0.25, 0.35, 0.57, and 1 MPa. After performing the test with the highest pressure, a last measurement was performed with the lowest value of the average contact pressure, in order to verify the eventual modiﬁcations of the interface (e.g., roughness plastic deformation) that could modify the natural frequency of the investigated mode. Figure5 shows the frequency response functions computed for each tested height of the samples, highlighting how the natural frequency of the investigated mode (at about 400–450 Hz) shifted at the lower frequencies, with the increase of the sample height. Such a trend was due to the decrease of the sample bulk stiﬀness, when its length increased. The frequency peak at about 150 Hz was related to a structural mode of the set-up and, as expected, it did not change as a function of the heights of the tested sample. After computing the FRFs as a function of the average contact pressure, sample length and sample roughness (as in Figure5), the frequency evolution of the investigated mode is plotted in Figure6. A decrease in the natural frequency as a function of the contact pressure was observed for both Ra = 1 µm and Ra = 6 µm. The decrease in frequency was due to the increase of the added mass weight needed for varying the contact pressure in the experiments. Lubricants 2019, 7, x FOR PEER REVIEW 9 of 17

Figure 6. Natural frequency of the investigated mode as a function of the average contact pressure for threethree didifferentﬀerent lengthslengths ofof thethe samplesample andand forforRa Ra= = 11 µµmm ((lowerlower)) andand RaRa= = 66 µµmm ((upperupper).).

3.2. Contact Stiffness Identification Figure 7 shows a comparison of the results related to the numerical and experimental tests in terms of the natural frequency of the investigated system mode. The numerical value of the normal contact stiffness, introduced in the contact model by penalty formulation, allowed for obtaining a quantitative agreement between the experiments and the numerical results, in the case of an average contact pressure of 0.14 MPa. The identified contact stiffness was Kc = 2.7 × 1011 (Pa/m) and Kc = 3 × 1011 (Pa/m) for a surface roughness of Ra = 1 µm and Ra = 6 µm, respectively. The use of samples of different lengths was adopted to verify the decoupling between the respective contributions of the bulk and the interface stiffness, in the overall stiffness of the investigation mode of vibration. In fact, when the length of the sample increased, only the stiffness of the bulk decreased. The fact that the numerical values of the natural frequencies in Figure 7 agreed the experimental values, both in magnitude and in direction (with respect to the sample length), meant that the contribution from the bulk and interface stiffness were correctly decoupled. While this result could be considered trivial, as the bulk material properties of the PMMA were known, it could be useful to provide such validation when dealing with brake lining materials [29], for which the material properties are affected by large incertitude. The results were also compared with the ones obtained numerically when using the contact stiffness calculated by the numerical software as default, i.e., Kc = 6 × 1013 (Pa/m). The comparison with the experiment highlighted a crucial difference in terms of the natural frequency of the considered system mode, confirming the unrealistic values of the contact stiffness in the commercial codes, which was overestimated and therefore led to an overestimation of the mode frequency (Figure 7). In fact, the contact stiffness calculated by the software did not account for the interface features, such as roughness or presence of a third body.

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3.2. Contact Stiffness Identification Figure7 shows a comparison of the results related to the numerical and experimental tests in terms of the natural frequency of the investigated system mode. The numerical value of the normal contact stiffness, introduced in the contact model by penalty formulation, allowed for obtaining a quantitative agreement between the experiments and the numerical results, in the case of an average contact pressure of 0.14 MPa. The identified contact stiffness was Kc = 2.7 1011 (Pa/m) and Kc = 3 1011 (Pa/m) for a × × surface roughness of Ra = 1 µm and Ra = 6 µm, respectively. Lubricants 2019, 7, x FOR PEER REVIEW 10 of 17

Figure 7.7. ComparisonComparison betweenbetween numerical numerical and and experimental experimental results results for for surface surface roughness roughness of Ra of= Ra6 µ=m 6 (µmupper (upper) and) andRa = Ra1 µ= m1µm (lower (lower)—average)—average contact contact pressure pressure of of P =P =0.14 0.14 MPa. MPa. Grey Grey pointspoints inin thethe plotsplots represent thethe numerical numerical results results in termsin terms of the of naturalthe natural frequency frequency obtained obtained with default with contactdefault sticontactffness 13 13 ofstiffnessKc = 6 of 10Kc = 6(Pa × 10/m). (Pa/m). × TheThen, use the of contact samples stiffness of different was lengths identified, was adoptedthrough tocomparison verify the decoupling with the experiments between the for respective both Ra contributions= 1 µm and Ra of the= 6 bulkµm, andas a thefunction interface of the stiff ness,average inthe contact overall pressure. stiffness In of Figure the investigation 8 the trend mode of the of vibration.contact stiffness In fact, as when a function the length of the of theaverage sample contact increased, pressure only for the both stiffness the ofsurface the bulk roughness, decreased. Ra The = 1 factµm and that Ra the = numerical 6 µm is presented. values of The the respective natural frequencies error bars in were Figure calculated7 agreed on the three experimental different samples values, bothfor each in magnitude test condition. and The in direction increase (within contact respect stiffness to the as sample a function length), of the meant contact that pressure the contribution was also fromconfirmed the bulk from and the interface results obtained stiffness wereby Xicorrectly Shi and Polycarpou decoupled. in While [43], thisfor a result dry contact could be system. considered trivial,The as results the bulk highlighted material properties how the ofcontact the PMMA stiffne weress increased known, itwith could the be increase useful toin provide the average such validationcontact pressure, when dealing for both with roughness brake lining parameters, materials [29 as], forfound which in thethe material literature properties by analytical are affected and bysimplified large incertitude. numerical models [30,40] and by ultrasonic experiments [44,45]. The estimated stiffness rangedThe between results 2.7 were × 10 also11 (Pa/m) compared and 6.1 with × 10 the11 (Pa/m), ones obtained when increasing numerically the contact when usingpressure the between contact 13 sti0.1ff nessMPa calculatedand 1 MPa. by The the increase numerical in softwarecontact stiffness as default, as i.e.,a functionKc = 6 of10 the (Pacontact/m). Thepressure comparison was in × withagreement the experiment with the theoretical highlighted and a crucial numerical difference results in presented terms of the in natural[46], in frequencywhich contact of the stiffness considered was systemstudied mode, by using confirming an elastic the contact unrealistic model values between of the a smooth contact stiplaneffness and in a the rough commercial surface. codes, which was overestimatedA difference in and terms therefore of the led co tontact an overestimation stiffness was ofhighlighted the mode frequency(Figure 8) (Figureas a function7). In fact, of thethe different surface roughness. A lower value of the surface roughness resulted in a higher value of the contact stiffness (black curve in Figure 8), within the whole range of the tested contact pressures. The trend in Figure 8 was in agreement with the power–law relation between the normal contact stiffness and pressure, theoretically derived in [47,48]. Moreover, the effect of roughness on contact stiffness followed the same behavior as found by the analytical and numerical contact model presented in [47].

Lubricants 2019, 7, 56 10 of 17 contact stiffness calculated by the software did not account for the interface features, such as roughness or presence of a third body. Then, the contact stiffness was identified, through comparison with the experiments for both Ra = 1 µm and Ra = 6 µm, as a function of the average contact pressure. In Figure8 the trend of the contact stiffness as a function of the average contact pressure for both the surface roughness, Ra = 1 µm and Ra = 6 µm is presented. The respective error bars were calculated on three different samples for each test condition. The increase in contact stiffness as a function of the contact pressure was also confirmedLubricants 2019 from, 7, x the FOR results PEER REVIEW obtained by Xi Shi and Polycarpou in [43], for a dry contact system.11 of 17

FigureFigure 8. 8.Normal Normal contact contact sti ffstiffnessness and and respective respective error error bars, asbars, a function as a function of the average of the contactaverage pressure contact forpressure surface for roughness surface roughness of Ra = 1 µ mof (Rared = curve1 µm) ( andred curveRa = 6) µandm (Rablack = 6 curveµm (black) in the curve sticking) in the condition. sticking condition. The results highlighted how the contact stiffness increased with the increase in the average contact4. Contact pressure, Stiffness for both Estimation roughness in parameters,the Sliding Condition as found in the literature by analytical and simplified numerical models [30,40] and by ultrasonic experiments [44,45]. The estimated stiffness ranged In several applications, the numerical simulations were aimed to reproduce contact in the sliding between 2.7 1011 (Pa/m) and 6.1 1011 (Pa/m), when increasing the contact pressure between 0.1 MPa conditions.× Consequently, the estimation× of the contact stiffness between bodies in relative motion and 1 MPa. The increase in contact stiffness as a function of the contact pressure was in agreement represents a significant improvement in numerical and analytical modeling [49] of contact frictional with the theoretical and numerical results presented in [46], in which contact stiffness was studied by systems. To achieve such a result, it was necessary to define an experimental procedure that allowed using an elastic contact model between a smooth plane and a rough surface. for obtaining consistent and reproducible results. A difference in terms of the contact stiffness was highlighted (Figure8) as a function of the To this purpose, the experimental tests were conducted according to the flow chart in Figure 9; different surface roughness. A lower value of the surface roughness resulted in a higher value of the the procedure is described below: contact stiffness (black curve in Figure8), within the whole range of the tested contact pressures. The First (at time 1 s in Figure 9), the system was excited in the sticking condition (no rotation trend in Figure8 was in agreement with the power–law relation between the normal contact sti ffness imposed on the disc), using the instrumented hammer with a single impact on the guide bar, while and pressure, theoretically derived in [47,48]. Moreover, the effect of roughness on contact stiffness recording the acceleration of the system; this preliminary sticking condition measurement allowed to followed the same behavior as found by the analytical and numerical contact model presented in [47]. get a reference test for comparing the subsequent tests performed in the sticking and sliding 4.conditions. Contact Sti Thenffness (at Estimation time 5 s in inFigure the Sliding 9), the motor Condition was activated, rotating the disc at a speed of 1.5 revolutions per second (940 mm/s of average sliding velocity), while the sample stayed fixed thanks to anIn edge several on applications,the support. theImpact numerical excitations simulations were provided were aimed during to reproduce the sliding, contact in order in theto perform sliding conditions.the identification Consequently, of the contact the estimation stiffness of in the the contact sliding sti condition.ffness between The average bodies infriction relative coefficient motion representsbetween the a significant PMMA specimen improvement and the in numericaldisc, measured and analyticalat the tested modeling sliding [velocity,49] of contact was 0.37. frictional After systems.80 s, the Tomotor achieve was suchdeactivated, a result, arresting it was necessary the moti toon define of the an disc. experimental The system procedure was then thatexcited allowed again forby obtainingthe instrumented consistent hammer, and reproducible performing results. a new measurement in the sticking condition without modifyingTo this purpose,the contact the (without experimental removing tests the were sample conducted). The previous according steps to the were flow repeated chart in to Figure reach9 ;at theleast procedure 240 m of is the described global sliding below: distance, with successive measurements in the sliding and sticking conditions, in order to follow the evolution of the contact stiffness, due to the evolution of the interface with the sliding distance. At the beginning and at the end of each campaign, the specimen surfaces were observed (Figure 10) and measured with a profilometer (Figure 11), in order to evaluate the modifications in the roughness profile and surface topography, between the beginning (initial contact surface) and the end of the sliding test (used contact surface).

Lubricants 2019, 7, 56 11 of 17 Lubricants 2019, 7, x FOR PEER REVIEW 12 of 17

FigureFigure 9. 9. FromFrom top top to to bottom—force bottom—force signal signal and and acceleration acceleration response; response; spectrum spectrum related related to to the the sticking sticking testtest 01 01 ( (blackblack curve curve),), sticking sticking test test 02 ( grey curve ), and sliding test at time t = 2828 s s ( redred curve curve).). Sample Sample lengthlength H H = =1010 mm, mm, roughness roughness parameter parameter Ra =Ra 6 µm,= 6 andµm, imposed and imposed average average contact contactpressure pressure of P = 0.35 of MPa.P = 0.35 Hammer MPa. impacts Hammer and impacts the corresponding and the corresponding FRFs were performed FRFs were during performed the sliding during evolution the sliding of theevolution interface. of the interface.

TheFirst results (at time from 1 s in a Figuretest, performed9), the system with was a PMMA excited inmaterial the sticking sample condition with length (no rotation L = 10 imposedmm, an imposedon the disc), average using contact the instrumented pressure of hammer 0.35 MPa, with and a a single surface impact roughness on the Ra guide = 6 bar,µm, whileis shown recording in Figure the 9.acceleration The figure of reports the system; the signal this preliminary of the force sticking due to condition the hammer measurement excitation, allowed the time to get acceleration a reference response,test for comparing and the spectra the subsequent computed tests at different performed phases in theof the sticking test. and sliding conditions. Then (at timeThe 5 s inblack Figure curve9), the was motor the response was activated, spectrum rotating related the to disc the atfirst a speed impulsive of 1.5 excitation revolutions of perthe secondsystem for(940 a mmsticking/s of averagecondition sliding measurement velocity), while(no rotation the sample imposed stayed to fixedthe disc). thanks On to the an other edge onhand, the support.the red curveImpact was excitations the spectrum were of provided the system during response the sliding,for an impulsive in order toexcitation perform given the identification during the rotation of the ofcontact the disc, stiff i.e.,ness in in the the sliding sliding condition, condition. at The about average 38 s frictionafter the coe beginningfficient between of the test. the PMMAThe comparison specimen ofand the the two disc, spectrum measured curves at the highlighted tested sliding a slight velocity, decrease was 0.37. in the After investigated 80 s, the motor mode was frequency deactivated, and, therefore,arresting thea decrease motion in of the the related disc. The contact system stiffnes wass, then in the excited sliding again condition. by the Moreover, instrumented the response hammer, spectrumperforming computed a new measurement at t = 103 s (grey in the curve sticking in Figu conditionre 9), when without the disc modifying was stopped, the contact highlighted (without a slightremoving increase the sample).in the investigated The previous system steps frequency, were repeated resulting to reach in a at moderate least 240 increase m of the in global the contact sliding stiffness,distance, with successiverespect to the measurements one measured in the at slidingthe beginning and sticking of the conditions, test. Such ina behavior order to follow could thebe associatedevolution ofto thethe contact variation stiff ofness, the duetopography to the evolution of the contact of the interface interface, with in term the sliding of roughness distance. and a third-bodyAt the beginninglayer, which and could at the have end of produced each campaign, an evolution the specimen in stiffness surfaces throughout were observed the entire (Figure test, 10) bothand measuredin the sticking with and a profilometer sliding conditions. (Figure 11In), order in order to estimate to evaluate the contact the modifications stiffness during in the the roughness sliding conditionprofile and tests, surface when topography, considering between the numerical the beginning model presented (initial contact in Section surface) 2.4, the and contact the end stiffness of the ofsliding the surface test (used in relative contact motion surface). was varied to meet the recovered experimental frequencies. On the other hand, the contact stiffness of the fixed surface (contact between sample and guide bar) was set to the value estimated previously in the static condition.

Lubricants 2019, 7, x FOR PEER REVIEW 13 of 17

Figure 10 shows the evolution of the contact stiffness (between sample and disc), as a function of the sliding distance, both in the sticking and sliding conditions, for a PMMA sample with H = 10 mm, Ra = 1 µm, and a contact pressure of P = 0.35 MPa. During the whole test, the results highlighted as the contact stiffness estimated in the sliding condition (orange curve in Figure 10) were always found to be lower than the one in the sticking condition (blue curve in Figure 10). This difference could be attributed to the lower local stiffness at the interface between the two materials when the asperities were not in the static adhesion, but rather slid on each other, bringing to a softening of the interface. Moreover, in both cases, the contact stiffness did not show a constant and stable behavior during the entire sliding test and its evolution was characterized by slow oscillations. This variation of the contact stiffness was a consequence of the evolution and rheology of the contact interface (see images in Figure 10). In fact, the continuous evolution of the surface topography and the third-body flows could bring to the accumulation and expulsion of third-body particles, which consequently affected the macroscopic stiffness of the interface. The effect of the third body on the contact stiffness was initially investigated in [43] and it was found that wear debris could significantly decrease the contact Lubricants 2019, 7, 56 12 of 17 stiffness.

Figure 10. a Ra µ Figure 10. ((a)) EvolutionEvolution of of the the contact contact sti stiffnessffness (disc (disc/sample)/sample) for for the the sample sample with with H = H10 = mm, 10 mm,= Ra1 =m, 1 and average contact pressure of 0.35 MPa. Blue dots are associated to the contact stiffness values µm, and average contact pressure of 0.35 MPa. Blue dots are associated to the contact stiffness values estimated in sticking condition while orange dots are associated to the contact stiffness values estimated estimated in sticking condition while orange dots are associated to the contact stiffness values in sliding condition; (b) Topography of the initial contact surface of the sample; (c) Topography of the estimated in sliding condition; (b) Topography of the initial contact surface of the sample; (c) contact surface of the sample at the end of the measurement campaign. Topography of the contact surface of the sample at the end of the measurement campaign. The results from a test, performed with a PMMA material sample with length L = 10 mm, Figure 11a shows a comparison of the contact stiffness evolution in the sliding conditions for an imposed average contact pressure of 0.35 MPa, and a surface roughness Ra = 6 µm, is shown in two values of the initial roughness, Ra = 6 µm and Ra = 1 µm, for a sample of H = 10 mm and an Figure9. The figure reports the signal of the force due to the hammer excitation, the time acceleration response, and the spectra computed at different phases of the test. The black curve was the response spectrum related to the first impulsive excitation of the system for a sticking condition measurement (no rotation imposed to the disc). On the other hand, the red curve was the spectrum of the system response for an impulsive excitation given during the rotation of the disc, i.e., in the sliding condition, at about 38 s after the beginning of the test. The comparison of the two spectrum curves highlighted a slight decrease in the investigated mode frequency and, therefore, a decrease in the related contact stiffness, in the sliding condition. Moreover, the response spectrum computed at t = 103 s (grey curve in Figure9), when the disc was stopped, highlighted a slight increase in the investigated system frequency, resulting in a moderate increase in the contact stiffness, with respect to the one measured at the beginning of the test. Such a behavior could be associated to the variation of the topography of the contact interface, in term of roughness and a third-body layer, which could have produced an evolution in stiffness throughout the entire test, both in the sticking and sliding conditions. In order to estimate the contact stiffness during the sliding condition tests, when considering the numerical model presented in Section 2.4, the contact stiffness of the surface in relative motion was varied to meet the recovered experimental frequencies. On the other hand, the contact Lubricants 2019, 7, x FOR PEER REVIEW 14 of 17

average contact pressure of P = 0.35 MPa. The results confirmed the trend of the contact stiffness recovered in the sliding condition, for both roughness values. It could be noted that, while the contact stiffness for Ra = 6 µm was lower at the beginning of the sliding, it became larger than that of the surface with Ra = 1 µm, after a run-in of the surfaces. In fact, at the beginning of the test, the lower stiffness was obviously a consequence of the higher roughness; afterwards, the smoothening of the asperities and filling of the roughness by the third-body (Figure 11c), made the interface more compact, bringing to a higher contact stiffness. LubricantsThis2019 conclusion, 7, 56 was supported by the comparison between Figure 11b and Figure 11c,13 which of 17 showed a smoother profile with the accumulation of third body, within the roughness grooves, for the surface sample at the end of the test. On the contrary, the lower roughness of the sample with Ra sti=ﬀ 1ness µm, ofdid the not ﬁxed allow surface for a consistent (contact between accumulation sample of and the guidethird body, bar) was which set towas the mainly value expulsed estimated at previouslythe border in of the the static contact condition. (Figure 10c).

FigureFigure 11. 11.(a )(a Evolution) Evolution of theof contactthe contact stiffness stiffness for the for sample the sample with H with= 10 mm,H = average10 mm, contact average pressure contact ofpressure 0.35 MPa, of and0.35 for MPa, both and roughness for both values, roughnessRa = 6values,µm and RaRa = =6 µm1 µm. and Orange Ra = 1 points µm. Orange are associated points toare theassociated contact sti toff theness contact values stiffness estimated values in the estimated sliding condition in the sliding for Ra condition= 1 µm. Green for Ra points = 1 µm. are Green associated points toare the associated contact sti fftoness the valuescontact estimated stiffness invalu thees sliding estimated condition in the for slidingRa = 6conditionµm; (b) Topography for Ra = 6 µm; of the (b) initialTopography contact surface,of the initial with thecontact associated surface, profile, with the for Raassociated= 6 µm; profile, (c) Topography for Ra = of6 µm; the contact (c) Topography surface, withof the the contact associated surface, profile, with at the the associated end of the profile, sliding at test. the end of the sliding test.

Figure 10 shows the evolution of the contact stiffness (between sample and disc), as a function of the sliding distance, both in the sticking and sliding conditions, for a PMMA sample with H = 10 mm, Ra = 1 µm, and a contact pressure of P = 0.35 MPa. During the whole test, the results highlighted as the contact stiffness estimated in the sliding condition (orange curve in Figure 10) were always found to be lower than the one in the sticking condition (blue curve in Figure 10). This difference could be attributed to the lower local stiffness at the interface between the two materials when the asperities were not in the static adhesion, but rather slid on each other, bringing to a softening of the interface. Lubricants 2019, 7, 56 14 of 17

Moreover, in both cases, the contact stiffness did not show a constant and stable behavior during the entire sliding test and its evolution was characterized by slow oscillations. This variation of the contact stiffness was a consequence of the evolution and rheology of the contact interface (see images in Figure 10). In fact, the continuous evolution of the surface topography and the third-body flows could bring to the accumulation and expulsion of third-body particles, which consequently affected the macroscopic stiffness of the interface. The effect of the third body on the contact stiffness was initially investigated in [43] and it was found that wear debris could significantly decrease the contact stiffness. Figure 11a shows a comparison of the contact stiffness evolution in the sliding conditions for two values of the initial roughness, Ra = 6 µm and Ra = 1 µm, for a sample of H = 10 mm and an average contact pressure of P = 0.35 MPa. The results confirmed the trend of the contact stiffness recovered in the sliding condition, for both roughness values. It could be noted that, while the contact stiffness for Ra = 6 µm was lower at the beginning of the sliding, it became larger than that of the surface with Ra = 1 µm, after a run-in of the surfaces. In fact, at the beginning of the test, the lower stiffness was obviously a consequence of the higher roughness; afterwards, the smoothening of the asperities and filling of the roughness by the third-body (Figure 11c), made the interface more compact, bringing to a higher contact stiffness. This conclusion was supported by the comparison between Figures 11b and 11c, which showed a smoother profile with the accumulation of third body, within the roughness grooves, for the surface sample at the end of the test. On the contrary, the lower roughness of the sample with Ra = 1 µm, did not allow for a consistent accumulation of the third body, which was mainly expulsed at the border of the contact (Figure 10c).

5. Conclusions In mechanical systems including contact interfaces, the interface parameters play a key role on the static and dynamic response of the system. Such parameters could deeply affect the local contact behavior and the global response of the system. To elaborate, in real frictional interfaces, the effective contact stiffness is a function of several operational parameters (roughness, contact pressure, third body rheology, surface topography) and it can affect both the local contact behavior (stress and strain at the contact interface) and the system global dynamics. In this study, a newer approach that combines numerical and experimental analysis was developed, in order to estimate the contact stiffness, both in sticking and sliding conditions, accounting for the evolution of the contact surface (topography, third body, etc.). An experimental setup was developed to perform experimental campaigns dedicated at the contact stiffness estimation. On the other hand, a numerical finite element model, accounting for both the dynamics of the setup and the interface stiffness, was developed and then updated by preliminary dynamic tests. The methodology for the estimation of more suitable values of the contact stiffness was validated here on the PMMA material samples. The quantitative estimation of contact stiffness, as a function of the average contact pressure and surface roughness was performed, showing how higher contact pressure and finer surface roughness led to an increase in the interface stiffness. The retrieved stiffness was found to be much lower than the one calculated as a default value, through the numerical commercial software, which did not account for the local topography and rheology of the contact. This overestimation of the numerical stiffness could lead to large errors in the respective numerical simulations, which could be reduced by the proposed identification of the effective stiffness. For the same reason, contact stiffness in the sliding conditions was estimated to account for its reduction when the interfaces were in the sliding conditions. The effect of the contact interface evolution (profile evolution and third body flows) on the contact stiffness was highlighted for both the sticking and the sliding conditions for the two values of the initial roughness. The presented results were in good agreement with the results recovered by the analytical/numerical models presented in [43,46–48]. Lubricants 2019, 7, 56 15 of 17

These results could provide significant improvement in the numerical modeling of frictional systems. In particular, finite element models, for both static and dynamic contact simulations, could be improved by introducing the effect of the interface stiffness, when performing analyses such as CEA and nonlinear transient analysis for brake squeal prediction.

Author Contributions: All authors conceived the experimental setup and the numerical model. All authors contributed to the interpretation of the results and provided critical feedback and helped shape the research, analysis and manuscript. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

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